Question: Solve for $x$ : $ 5|x + 4| - 8 = -1|x + 4| + 2 $
Add $ {1|x + 4|} $ to both sides: $ \begin{eqnarray} 5|x + 4| - 8 &=& -1|x + 4| + 2 \\ \\ { + 1|x + 4|} && { + 1|x + 4|} \\ \\ 6|x + 4| - 8 &=& 2 \end{eqnarray} $ Add ${8}$ to both sides: $ \begin{eqnarray} 6|x + 4| - 8 &=& 2 \\ \\ { + 8} &=& { + 8} \\ \\ 6|x + 4| &=& 10 \end{eqnarray} $ Divide both sides by ${6}$ $ \dfrac{6|x + 4|} {{6}} = \dfrac{10} {{6}} $ Simplify: $ |x + 4| = \dfrac{5}{3}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 4 = -\dfrac{5}{3} $ or $ x + 4 = \dfrac{5}{3} $ Solve for the solution where $x + 4$ is negative: $ x + 4 = -\dfrac{5}{3} $ Subtract ${4}$ from both sides: $ \begin{eqnarray} x + 4 &=& -\dfrac{5}{3} \\ \\ {- 4} && {- 4} \\ \\ x &=& -\dfrac{5}{3} - 4 \end{eqnarray} $ Change the ${ - 4}$ to an equivalent fraction with a denominator of $3$ $ x = - \dfrac{5}{3} {- \dfrac{12}{3}} $ $ x = -\dfrac{17}{3} $ Then calculate the solution where $x + 4$ is positive: $ x + 4 = \dfrac{5}{3} $ Subtract ${4}$ from both sides: $ \begin{eqnarray} x + 4 &=& \dfrac{5}{3} \\ \\ {- 4} && {- 4} \\ \\ x &=& \dfrac{5}{3} - 4 \end{eqnarray} $ Change the ${ - 4}$ to an equivalent fraction with a denominator of $3$ $ x = \dfrac{5}{3} {- \dfrac{12}{3}} $ $ x = -\dfrac{7}{3} $ Thus, the correct answer is $x = -\dfrac{17}{3} $ or $x = -\dfrac{7}{3} $.